Objective: What is the speed of a Constant Motion Vehicle (CMV) Hypothesis: A CMV moves approximately 0.2 cm/s.
Position Time Data for a CMV:
Graph:
Analysis:
After graphing the data, it was revealed that the graph is linear, meaning that the CMV did move at a constant speed.
Graph Equation: y = 11.582x + 0.99574m
Discussion Questions: 1. Why is the slope of the position-time graph equivalent to average velocity? The slope is represented by change in position over change in time, and the velocity is represented as change in distance, or position, over change in time. The same data was used in the same spot when finding both, so therefore the answers would have to be the same. 2. Why is it average velocity and not instantaneous velocity? What assumptions are we making? Average velocity is used over a particular time interval to find its displacement during the time interval. Instantaneous velocity is used for a range of speeds and from our results; we can conclude that ours only had one constant speed 3. Why was it okay to set the y-intercept equal to zero? It was okay to set the y-intercept equal to zero because we knew that it was the starting point and that no time had passed. We can only do this when we know that the starting point is zero, if we do not know the starting point or if it was possible that it was less than zero then we would not have been able to do it. 4. What is the meaning of the R2 value? It is used as an evaluation of how accurate the points are to the trend-line on the graph. It is a percentage of how well we measured the dots on the tape. 5. If you were to add the graph of another CMV that moved more slowly on the same axes as your current graph, how would you expect it to lie relative to yours? My CMV seemed to be very slow compared to others, so therefore I would find it to be pretty similar. Both graphs would have similar data and the line of best fit would be very close to mine because of the slow speed.
CONCLUSION: Before performing the lab, I believed that a CMV moves roughly around 0.2 m/s. In my lab, I had a very slow one battery CMV, so mine moved around 11.582cm/s. This part of my hypothesis was mostly inaccurate, because I should have estimated something lower. Before, I also said that the most precise distances can be measured is to give a starting and ending point, but the real answer is extended the decimal place, so that the measurement is more accurate. The most accurate part of my hypothesis is that a position-time graph tells you the position at a certain time. After looking at the tape, I realized that all the dots were pretty much at the same rate and stayed a constant speed. When measuring each distance, there were many opportunities for errors to be made. For example, the positioning of the ruler could have easily slipped or slide a certain way without someone noticing, and this would cause the wrong data to be accounted for. Another source of error that could have contributed to an inaccuracy is the decimal points. It is hard to tell what the exact measurement is on a ruler, so there could have been many decimal points that were off. To minimize these issues, one can use a measuring tape instead of a ruler so that it will have a lesser chance of sliding. New batteries could also be added to the CMV’s to make sure everything is consistent.
Lesson 1: Describing Motion with Words
Homework Questions for Lesson 1(9/8/11) 1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
In this reading and our class discussions, we talked about how distance and displacement relate to one another and what they are both used for. Even though I already knew that distance referred to "how much ground an object has covered" during its motion and that displacement refers to "how far out of place an object is", it still helped me remember the important parts of our class discussions. I remember talking about how in displacement things can be cancelled out so in the end the displacement can be equal to 0. Another thing that I already knew from class was that because speed is a scalar quantity it does not require direction. This is unlike velocity because it is a vector quantity, meaning that it requires both direction and a numerical measurement. 2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
I was a little confused on speed vs. velocity because I never knew or was taught that there was a difference between the two. I had always just thought they both meant the same thing and both deal with how fast an object is going. The reading helped me to understand that speed is specifically "how fast an object is moving", while velocity refers to "the rate at which an object changes its position". 3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
I pretty much understood everything that the reading covered. I think that it had a lot of detail for each topic, which helped me learn and understand what each was about. 4. What (specifically) did you read that was not gone over during class today?
I did not remember ever going over scalars and vectors, but after reading about them I have a pretty good understanding of what each is now.
After reading the section, I realized that a scalar quantity is just the magnitude alone, or just the numerical value. The reading also helped me realize that vectors require both the magnitude and a direction.
Class Notes on Lesson 1:
Displacement
change in position
Velocity
how fast you make that change
Average Speed
total speed/total time
can also be found if you know two speeds
Constant speed
not changing
Instantaneous speed
always the same
should be really small
- All of these three things use the same equation V= (total distance/total time)
TYPES OF MOTION
1) at rest
in the same place, standing still
2) constant speed
speed is not changing
3) increasing speed
going faster
4) decreasing speed
going slower
-Both increasing and decreasing speed represents acceleration (changing speeds in general)
Motion Diagrams:
show direction of velocity
if at rest, v=0, a=0
Constant Speed:
---> ---> ---> a=0 OR <--- <--- <--- a=0
(each arrow representing velocity) a=0
- all the arrows are same size because the speed is remaining the same
Decreasing Speed:
------------> ------> ---->
a= <------ (because it is negative)
Ticker Tape Diagrams:
More used for numerical
Look under Lesson 2 for more info & examples
Ticker Tape Examples:
Ticker Tape vs. Motion Diagram:
- Motion Diagram has very precise measurements over ticker tape diagram
- can't see direction in ticker tape
Signs are arbitrary (made up, someone just said so)
- to change it is it allowed, but you have to label it when you do it so others know
- all you do is look at the direction in which it points and that tells you what the sign is (+ or -)
up and to the right = positive
down and to the left = negative
- Kinematics is the study of motion
Velocity
if you're going in a circle and end up where you started like running on a track then the v=0.
Lesson 2: Describing Motion with Diagrams
Introduction to Diagrams:
Goal: to be able to represent motion in a variety of different ways.
Ticker Tape Diagrams:
Ticker tape analysis: a long tape is attached to a moving object and then threaded through a device that places a tick upon regular intervals of time.
The distance between dots on a ticker tape represents the object's position change during that time interval. A large distance between dots indicates that the object was moving fast during that time interval. A small distance between dots means the object was moving slow during that time interval.
The analysis of a ticker tape diagram will also reveal if the object is moving with a constant velocity or accelerating.
A changing distance between dots indicates a changing velocity and thus an acceleration.
A constant distance between dots represents a constant velocity and therefore no acceleration.
Vector Diagrams:
Diagrams that depict the direction and relative magnitude of a vector quantity by a vector arrow.
The magnitude of a vector quantity is represented by the size of the vector arrow.
If the size of the arrow in each consecutive frame of the vector diagram is the same, then the magnitude of that vector is constant.
Be familiar with the concept of using a vector arrow to represent the direction and relative size of a quantity
RULE OF THUMB:
Homework Question for Lesson 2: (9/9/11)
After reading the material, answer the following questions:
1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
I already knew about and understood a lot about the ticker tape diagram because of our lab. We talked about in class how to tell when it was speeding up or slowing down, so when reading about it I already knew about the concept. I knew about the specific spacing between each dot and what each placement represented.
Also in class, I understood the concept of what the different arrow sizes meant on a vector diagram. I didn't have any confusion on if an object was accelerating or was at a constant speed.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
While I was reading, I started to realize the difference between velocity and acceleration, which I was a little confused on during class. But, after reading about it and learning about the "rule of thumb" from the animation, I feel that I am not as unclear about the subject now.
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
I think everything I read I pretty much understood. Some things I was a little confused on going into the reading, but after learning about certain subjects a second time, I believe it helped me understand everything fully.
4. What (specifically) did you read that was not gone over during class today?
I believe everything in the reading was covered at one point during class.
Class Notes for Lesson 2: (9/12/11)
Important Equations: (The Big 5)
Kinematics example problem:
Vector Diagram:
- only qualitative
ACTIVITY ON GRAPHICAL REPRESENTATIONS OF EQUILIBRIUM
9/12/11 GRAPHS:
No Motion
Constant Motion:
Fast Motion:
Slow Motion:
Discussion Questions:
1. How can you tell that there is no motion on a...
Position vs. time graph: it will just be a straight line
Velocity vs. time graph: it will be a straight line as well, but the velocity should be equal to zero
Acceleration vs. time graph: mostly be a straight line and the acceleration will be 0.
2. How can you tell that your motion is steady on a…
position vs. time graph: slope will be steady and constant
velocity vs. time graph: no slope, straight line
acceleration vs. time graph: no slope, straight line
3. How can you tell that your motion is fast vs. slow on a…
position vs. time graph: the distances between the times will be greater if it is going faster, and they will be closer if it is slower
velocity vs. time graph: just stay the same
acceleration vs. time graph: it will just stay at zero, because if it is constantly going fast or slow, then there is no acceleration
4. How can you tell that you changed direction on a…
position vs. time graph: the slopes of each line will change to it's opposite
velocity vs. time graph: they would just be opposite
acceleration vs. time graph: it only shows change in speed
5. What are the advantages of representing motion using a…
position vs. time graph: shows direction, slope, and time intervals
velocity vs. time graph: shows the increase or decrease of velocity from the slope
acceleration vs. time graph: shows the change in speed from the slope
6. What are the disadvantages of representing motion using a…
position vs. time graph: doesn't specifically tell us the direction or acceleration
velocity vs. time graph:
acceleration vs. time graph: only shows the changes in speed
7. Define the following:
No motion: when an object remains at rest resulting in everything being equal to zero.
Constant speed: the velocity remains the same throughout, while the acceleration remains at zero because there is no change in speed.
GRAPH PROBLEMS:
Speed is Constant Positive, then itdecreases, then it's at rest
Notes on Lesson 1-E: Acceleration
Acceleration: a vector quantity that is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.
Acceleration has to do with changing how fast an object is moving
Constant Acceleration: the velocity is changing by a constant amount each second.
If an object is changing its velocity -whether by a constant amount or a varying amount - then it is an accelerating object. And an object with a constant velocity is not accelerating.
Because acceleration is a vector quantity, that means it has direction associated with it.
The direction of the acceleration vector depends on two things:
whether the object is speeding up or slowing down
whether the object is moving in the + or - direction
RULE OF THUMB: If an object is slowing down, then its acceleration is in the opposite direction of its motion.
Homework Questions: 1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
ready knew that acceleration did not just mean an object is increasing speed. It is mainly just the rate at which an object is changing its velocity. Another thing that I already knew from class was that the direction of the acceleration vector depends on two things, whether the object is speeding up or slowing down and whether the object is moving in a positive or negative direction. 2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
During class, I was a little confused on the different units for average velocity. The reading helped me understand which units are for acceleration and what ones are for velocity. 3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
Everything in the reading was pretty clear. 4. What (specifically) did you read that was not gone over during class today?
I don't really remember going over anything on free-falling objects. The reading explained it clearly though and now I understand that the object would have a constant acceleration due to the fact that it is falling.
Accelerating on an Incline Lab:
Danielle & Jake - 9/14/11 Objectives:
What does a position-time graph for increasing speeds look like?
What information can be found from the graph?
Hypothesis:
A position-time graph for increasing speeds would have a constant positive slope, and form a diagonal line.
Procedure:
1) Gather the materials that are needed from inside the classroom.
2) Place the ramp on top of the Physics textbook so that there is an incline.
3) Put the ticker tape through the spark timer.
4) Place the spark timer at the top of the ramp and attach the end of the ticker tape to the cart.
5) Make sure that the spark timer is on 10 Hertz then turn it on.
6) Allow the cart to drop from the top of the ramp to the bottom, forming a downward incline.
7) Remove the ticker tape from the spark timer.
8) Using a measuring device, measure the distance between each dot.
9) After recording the data, place the cart at the bottom of the ramp and then push it up.
10) Repeat steps 7 & 8.
11) Graph & analyze the recorded data on a position-time graph for both inclines.
Analysis:
Increasing Down Incline Data:
Decreasing Up Incline Data:
GRAPH:
Analysis:
a) Interpret the equation of the line (slope, y-intercept) and the R2 value.
y= Ax2+Bx
y= distance, x=time
change in distance= 1/2at2 + vit
Accelerating:
Equation of the line: Δd= 15.797t2 + 6.5774t
R2= 0.99801
Decelerating:
Δd= 28.776t2 + 19.883t
R2= 0.9887
- We found that our cart had an acceleration of 31.594 m/s because we were able to substitute the data in.
- The r2 on our graph value shows that a large amount of our data fell within 1 unit of the target point.
Instantaneous speed:
halfway point = 61.37 m/s
end = 46.91 m/s.
Average speed:
19.64 m/s.
Discussion Questions: 1. What would your graph look like if the incline had been steeper?
If the incline had been steeper, then the curve would be more dramatic and steeper. This would be due to the fact that the points would be traveling the same distance just in a shorter amount of time. 2. What would your graph look like if the cart had been decreasing up the incline?
At first, the cart would start off at a fast rate, but then gradually curve the other way. This means that the slope would start off really steep, but then decrease, and this was what caused the average to decrease. At the halfway point, more speed was gained causing a greater accelerating graph. 3. Compare the instantaneous speed at the halfway point with the average speed of the entire trip.
This makes sense because our first couple of points were unique and off to the side because of the lower speed.
At the halfway point, more speed was gained causing a greater accelerating graph. Overall, the instantaneous speed reflects a further accelerating graph, while the average speed focuses more on the slow beginning. 4. Explain why the instantaneous speed is the slope of the tangent line. In other words, why does this make sense?
The instantaneous speed is the slope of the tangent line because the line has a constant slope. Due to the fact that the original line was curved, it allowed the tangent line to hit the midway point, which allows the instantaneous speed to be estimated. 5. Draw a v-t graph of the motion of the cart. Be as quantitative as possible.
Conclusion:
In our hypothesis, we said that a position-time graph for increasing speeds would have a constant positive slope, and form a diagonal line. This part of our hypothesis was wrong because after completing the experiment and graphing the data, there slope was not constant nor always positive, so therefore it was not a diagonal line that was formed. The resulting graph turned out to be more of a curve than we expected. The correct part of our hypothesis was when we said that we would be able to find the velocity from the graph. There were many errors sources of error that were made throughout this lab. One thing is that my partner and I kept messing up our graph. We couldn't figure out why ours looked so different, because we knew that the data was collected correctly. We had to record our data again by re-doing the experiment, until the problem was finally fixed. We just realized that the axis' were wrong. Another error could have been due to the fact that our measurements were not completely accurate because we didn't have the right technology that would tell us each accurate measure. The overall goal that I got from this lab was that on a position-time graph, when the cart is accelerating then it results in a curve.
Lesson 3: Describing Motion with Position vs. Time Graphs
The Meaning of Shape for a p-t Graph:
describes motion
A motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph.
Importance of Slope on this type of graph:
Reveals info on the velocity
Whatever characteristics the velocity has, the slope will exhibit the same (and vice versa).
If the velocity is constant, then the slope is constant (i.e., a straight line).
If the velocity is changing, then the slope is changing (i.e., a curved line).
If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right).
Determing the Slope:
Equation:
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. In other words,
Pick two points on the line and determine their coordinates.
Determine the difference in y-coordinates of these two points (rise).
Determine the difference in x-coordinates for these two points (run).
Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
During class, we talked about the difference between graphing positive and negative graphs. It was easy for me to understand while reading about it because I already grasped the concept while in class. I also already understood the shape of a position-time graph because of the motion sensor activity that we did.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
The reading helped me understand the right way to find the slope on a position time graph.
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
Everything I read was pretty clear.
4. What (specifically) did you read that was not gone over during class today?
I believe that everything I read was gone over.
Lesson 4: Describing Motion with Velocity vs. Time Graphs
The Meaning of Shape for a v-t Graph: Velocity vs. Time Graphs:
both used to describe motion
the motion depends on the specific shape and slope of the lines on the graphs
The slope of the line on a velocity-time graph reveals useful information about the acceleration of the object.
When you speed up and increase speed, the numbers increase and vice versa
The Meaning of Slope for a v-t graph:
In this situation above, the slope of the line is equal to the acceleration of the velocity-time graph.
However, a car with a changing velocity will have an acceleration.
Relating the Shape to the Motion:
In a velocity-time graph, if the acceleration is zero, then the velocity-time graph is a horizontal line - having a slope of zero.
If the acceleration is positive, then the line is an upward sloping line - having a positive slope.
If the acceleration is negative, then the velocity-time graph is a downward sloping line - having a negative slope.
If the acceleration is great, then the line slopes up steeply - having a large slope.
Determining the Slope on a v-t graph:
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. A method for carrying out the calculation is
Pick two points on the line and determine their coordinates.
Determine the difference in y-coordinates for these two points (rise).
Determine the difference in x-coordinates for these two points (run).
Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
Determining the area on a v-t graph:
For v-t graphs, the area bound by the line and the axes represents the displacement.
\
Calculating the Area of a Rectangle:
Calculating the Area of a Triangle:
Calculating the Area of Trapezoid:
Alternative Methods for Trapezoids: Homework Questions: 1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
I already understand how to find slope from math. I also already knew how to find the area of a triangle, trapezoid, and a rectangle. 2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
I remember us talking a little bit about the area of a shape in a v-t graph equaling the total displacement, the reading helped me understand it more than what I got out of it from the class discussion. 3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
Everything that I did not understand in class was pretty much covered in the reading. 4. What (specifically) did you read that was not gone over during class today?
I think that the same topics that were discussed in class were also covered in the reading.
A Crash Course in Velocity Lab (Part II)
9/21/11
Partners: Jake Aronson, Jessica Smith, & Michael Solimano
Objectives: Both algebraically and graphically, solve the following 2 problems. Then set up each situation and run trials to confirm your calculations. Procedure:
Answers to problems:
Answer to Problem #1: 174.31 cm (where they meet)
Answer to Problem #2: 69.376 cm (where they exceed)
Car #1
Fast
Yellow
CMV equation: y=28.274x
Car #2
Slow
Yellow
CMV equation: y=11.582x
Data for Trials:
Screen_shot_2011-09-21_at_11.55.12_AM.png
Discussion Questions: 1) Where would the cars meet if their speeds were exactly equal?
If both cars had the same speed then they would never meet. The first car would never slow down, so it would always be ahead of the second car, resulting in them never being in the same place at the same time. 2) Sketch position-time graphs to represent the catching up and crashing situations. Show the point where they are at the same place at the same time.
3) Sketch velocity-time graphs to represent the catching up situation. Is there any way to find the points when they are at the same place at the same time?
No, because on a velocity- time graph, position is not shown.
Conclusion:
Our lab group discovered that if both a slow, yellow CMV & a fast, yellow CMV started 600 cm apart, then they would "collide" after traveling about 217.5 cm. Our theoretical data showed that the CMV's would collide 174.31 cm. When we compared this data to our actual value, our %error was 24.95%. Our theoretical data also showed that the CMV's would catch up at 69.376 cm. After completing the lab, we found out that the faster CMV caught up to the slower CMV around 65.6 cm. We compared this data to our theoretical data, and found out that we had a percent of 11.514%. One of the main sources of error that could be taken into account was the fact that the CMV's did not travel in a straight line. This helped contribute to the wide range of numbers in some of our trials. To fix this error, we could have attached both the CMV's to a straight track, so that our numbers would be more accurate. Another reason why our percent error was so high could have been because the batteries could have been different from when we last used them.
Egg Drop Project
9.28.11 Partner: Dani Rubenstein
Final Project:
Description:
Our final device consisted of six cylinders holding the egg in the middle. In each of these cylinders, we placed a large amount of straws to serve as the main protective barrier for the egg. The straws also were what created the shape of the individual cylinders. We tried to build it as tight as we possible could, so therefore the egg would not fall out nor would it feel a large amount of pressure when it hit the ground.
Result:
Surprisingly after we dropped our egg it fully cracked and leaked everywhere.
Analysis:
In our first prototype, we built something that had very similar characteristics to our final one. They had the same shape, and both included a large amount of straws protecting each side. The only difference was that our first one was a little bigger, and weighed more. Even though our first protoptype was successful and kept the egg safe, we decided it would be better to minimize the size in the second prototype, so that it would weigh less. After testing out our second prototype, we learned that this shape and size was not efficient enough to keep the egg safe. In our final project, we decided to use the same design as the first one, except we cut it in half to reduce the weight. Although the first was successful, our final project did not support the egg enough causing it to crack everywhere. We learned that our project moved at a rate of 9.8 m/s2, which is very close to the average velocity of a free-falling object.
Calculations for a:
What would you do differently?
In order to make our project fully succeed, I believe that we should have built it in the shape of a cone, and placed the egg inside, surrounding it with newspaper. We also should have built some sort of parachute to slow our project down while it was in the air, and then it would have less force and impact when hitting the ground.
Lesson 5: Free Fall and the Acceleration of Gravity
Introduction to Free Fall:
A free falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects:
Free-falling objects do not encounter air resistance.
All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s (often approximated as 10 m/s/s for back-of-the-envelope calculations)
The position of the object at regular time intervals - say, every 0.1 second - is shown. The fact that the distance that the object travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. If an object travels downward and speeds up, then its acceleration is downward. The Acceleration of Gravity The acceleration of gravity
9.8 m/s/s
The acceleration for any object moving under the sole influence of gravity.
- Acceleration is the rate at which an object changes its velocity. It is the ratio of velocity change to time between any two points in an object's path. To accelerate at 9.8 m/s/s means to change the velocity by 9.8 m/s each second.
If the velocity and time for a free-falling object being dropped from a position of rest were tabulated, then one would note the following pattern.
Time (s)
Velocity (m/s)
0
0
1
- 9.8
2
- 19.6
3
- 29.4
4
- 39.2
5
- 49.0
Observe that the velocity-time data above reveal that the object's velocity is changing by 9.8 m/s each consecutive second. That is, the free-falling object has an acceleration of approximately 9.8 m/s/s.
Another way to represent this acceleration of 9.8 m/s/s is to add numbers to our dot diagram that we saw earlier in this lesson. The velocity of the ball is seen to increase as depicted in the diagram at the right. Representing Free Fall by Graphs: A position versus time graph for a free-falling object is shown below.
A velocity versus time graph for a free-falling object is shown below.
How Fast? & How far?
Free-falling objects are in a state of acceleration.
Specifically, they are accelerating at a rate of 9.8 m/s/s.
This is to say that the velocity of a free-falling object is changing by 9.8 m/s every second.
The formula for determining the velocity of a falling object after a time of t seconds is
vf = g * t
where g is the acceleration of gravity. The value for g on Earth is 9.8 m/s/s. The above equation can be used to calculate the velocity of the object after any given amount of time when dropped from rest. Example calculations for the velocity of a free-falling object after six and eight seconds are shown below.
Example Calculations: At t = 6 s
vf = (9.8 m/s2) * (6 s) = 58.8 m/s At t = 8 s
vf = (9.8 m/s2) * (8 s) = 78.4 m/s
The distance that a free-falling object has fallen from a position of rest is also dependent upon the time of fall. This distance can be computed by use of a formula; the distance fallen after a time of t seconds is given by the formula.
d = 0.5 * g * t2
where g is the acceleration of gravity (9.8 m/s/s on Earth). Example calculations for the distance fallen by a free-falling object after one and two seconds are shown below.
Example Calculations: At t = 1 s
d = (0.5) * (9.8 m/s2) * (1 s)2 = 4.9 m At t = 2 s
d = (0.5) * (9.8 m/s2) * (2 s)2 = 19.6 m At t = 5 s
d = (0.5) * (9.8 m/s2) * (5 s)2 = 123 m
(rounded from 122.5 m)
The diagram below (not drawn to scale) shows the results of several distance calculations for a free-falling object dropped from a position of rest.
The Big Misconception:
Nearly everyone has observed the difference in the rate of fall of a single piece of paper (or similar object) and a textbook. The two objects clearly travel to the ground at different rates - with the more massive book falling faster. The answer to the question (doesn't a more massive object accelerate at a greater rate than a less massive object?) is absolutely not!
Free-fall is the motion of objects that move under the sole influence of gravity; free-falling objects do not encounter air resistance. More massive objects will only fall faster if there is an appreciable amount of air resistance present.
The actual explanation of why all objects accelerate at the same rate involves the concepts of force and mass. The acceleration of an object is directly proportional to force and inversely proportional to mass. Increasing force tends to increase acceleration while increasing mass tends to decrease acceleration. Thus, the greater force on more massive objects is offset by the inverse influence of greater mass. Subsequently, all objects free fall at the same rate of acceleration, regardless of their mass.
.
Falling Object Lab
10/5/11 Partner: Jake Aronson
Purpose: What is the acceleration of a falling body?
Hypothesis:The acceleration due to gravity of a free-falling object is about -980 cm/s2.The v-t graph for this lab should be negative slant away from the origin. We will find the "g" value from the equation Ax2+bx, where the "A" will represent the "g" value.
Our Data: Class Data:
Analysis & Graphs: Position-Time Graph:
From this graph, we can see that our acceleration is 756.56 cm.s2. This is because the slope of our trend line is equal to the acceleration.We can calculate the acceleration by using the equation in which this graph uses, which is y=Ax2+Bx. The formula that we used in class for this graph was, y=vit+½at2. Being that the r2 is so close to 1 here, our data is almost perfect when compared to the trend line. Velocity-Time Graph:
This graph is graphed in the form of y=mx+b, or an equation of a line. The equation that we used in class for this graph is vf=vi+at. The difference between these two is that instead of multiplying the slope like we did in the first one, here we already have the acceleration given to us, 755.87 cm/s2. Just like in the position time graph above, our r2 value is very close to 1, so again we have an almost perfect match between our data and trend line.
Sample Calculations:
Discussion Questions:
1. Does the shape of your vt graph agree with the expected graph? Why or why not?
No, because on the expected graph the velocity line to be negative considering we were dropping something. But, the actually graph showed the complete opposite. This was because instead of plugging in the negative data, we used the positive data. On the expected graph, the velocity line was suppose to be on a negative slant away from the origin. Our graph showed a positive line, so they were completely different.
2. Does the shape of your xt graph agree with the expected graph? Why or why not?
Yes, both graphs agree with each other. Both graphs have a J-curve because of the increasing velocity and acceleration.
3. How do you results compare to that of the class? (Use percent difference to discuss quantitatively).
After graphing our data, we discovered that our slope was 755.28, and the class average was 839.417. After finding out both these numbers, we found that the %difference was 10.02, so we had about a 10% difference from the class.
4. Did the object acceleration uniformly? How do you know?
Yes, because in the v-t graph there is a constant slope.
5. What factor(s) would cause acceleration due to gravity to be higher than it should be? Lower than it should be?
One factor that could have caused acceleration due to gravity to be higher could have been the placement of our hands while we were dropping the weights. A factor that could have either cause the gravity to be higher or lower could have just been measurements because the tape was not sturdy the whole time. Conclusion:
Our data turned out to be very accurate and our hypothesis was not fully correct because of the fact that we predicted to have a negative velocity, but the graph turned out to have a positive slope. We thought that it would be negative because we were dropping something. Some causes of error that could have caused our results to be a little off form others could have been the friction that was caused from the ticker tape rubbing against the ticker tape timer. Another issue was when we were measuring our data. We could have made a mistake while doing this because the tape was not staying on the ground, and would keep coming up. To fix this, we could have been more cautious and taped it better so that it would be less likely to shift. We also could have dropped the weights from something other than our hands, so that no extra momentum would be caught.
A Crash Course in Velocity Lab: (Part 1)
9/7/11Partner: Jake Aronson
Table of Contents
Hypothesis: A CMV moves approximately 0.2 cm/s.
Position Time Data for a CMV:
Graph:
Analysis:
After graphing the data, it was revealed that the graph is linear, meaning that the CMV did move at a constant speed.
Graph Equation: y = 11.582x + 0.99574m
Discussion Questions:
1. Why is the slope of the position-time graph equivalent to average velocity?
The slope is represented by change in position over change in time, and the velocity is represented as change in distance, or position, over change in time. The same data was used in the same spot when finding both, so therefore the answers would have to be the same.
2. Why is it average velocity and not instantaneous velocity? What assumptions are we making?
Average velocity is used over a particular time interval to find its displacement during the time interval. Instantaneous velocity is used for a range of speeds and from our results; we can conclude that ours only had one constant speed
3. Why was it okay to set the y-intercept equal to zero?
It was okay to set the y-intercept equal to zero because we knew that it was the starting point and that no time had passed. We can only do this when we know that the starting point is zero, if we do not know the starting point or if it was possible that it was less than zero then we would not have been able to do it.
4. What is the meaning of the R2 value?
It is used as an evaluation of how accurate the points are to the trend-line on the graph. It is a percentage of how well we measured the dots on the tape.
5. If you were to add the graph of another CMV that moved more slowly on the same axes as your current graph, how would you expect it to lie relative to yours?
My CMV seemed to be very slow compared to others, so therefore I would find it to be pretty similar. Both graphs would have similar data and the line of best fit would be very close to mine because of the slow speed.
CONCLUSION:
Before performing the lab, I believed that a CMV moves roughly around 0.2 m/s. In my lab, I had a very slow one battery CMV, so mine moved around 11.582cm/s. This part of my hypothesis was mostly inaccurate, because I should have estimated something lower. Before, I also said that the most precise distances can be measured is to give a starting and ending point, but the real answer is extended the decimal place, so that the measurement is more accurate. The most accurate part of my hypothesis is that a position-time graph tells you the position at a certain time. After looking at the tape, I realized that all the dots were pretty much at the same rate and stayed a constant speed. When measuring each distance, there were many opportunities for errors to be made. For example, the positioning of the ruler could have easily slipped or slide a certain way without someone noticing, and this would cause the wrong data to be accounted for. Another source of error that could have contributed to an inaccuracy is the decimal points. It is hard to tell what the exact measurement is on a ruler, so there could have been many decimal points that were off. To minimize these issues, one can use a measuring tape instead of a ruler so that it will have a lesser chance of sliding. New batteries could also be added to the CMV’s to make sure everything is consistent.
Lesson 1: Describing Motion with Words
Homework Questions for Lesson 1(9/8/11)1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
In this reading and our class discussions, we talked about how distance and displacement relate to one another and what they are both used for. Even though I already knew that distance referred to "how much ground an object has covered" during its motion and that displacement refers to "how far out of place an object is", it still helped me remember the important parts of our class discussions. I remember talking about how in displacement things can be cancelled out so in the end the displacement can be equal to 0. Another thing that I already knew from class was that because speed is a scalar quantity it does not require direction. This is unlike velocity because it is a vector quantity, meaning that it requires both direction and a numerical measurement.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
I was a little confused on speed vs. velocity because I never knew or was taught that there was a difference between the two. I had always just thought they both meant the same thing and both deal with how fast an object is going. The reading helped me to understand that speed is specifically "how fast an object is moving", while velocity refers to "the rate at which an object changes its position".
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
I pretty much understood everything that the reading covered. I think that it had a lot of detail for each topic, which helped me learn and understand what each was about.
4. What (specifically) did you read that was not gone over during class today?
I did not remember ever going over scalars and vectors, but after reading about them I have a pretty good understanding of what each is now.
After reading the section, I realized that a scalar quantity is just the magnitude alone, or just the numerical value. The reading also helped me realize that vectors require both the magnitude and a direction.
Class Notes on Lesson 1:
Displacement
- change in position
VelocityAverage Speed
- total speed/total time
- can also be found if you know two speeds
Constant speed- not changing
Instantaneous speed- All of these three things use the same equation V= (total distance/total time)
TYPES OF MOTION
1) at rest
- in the same place, standing still
2) constant speed- speed is not changing
3) increasing speed- going faster
4) decreasing speed- going slower
-Both increasing and decreasing speed represents acceleration (changing speeds in general)Motion Diagrams:
Constant Speed:
---> ---> ---> a=0 OR <--- <--- <--- a=0
(each arrow representing velocity) a=0
- all the arrows are same size because the speed is remaining the same
Increasing Speed:
---> -------> ----------->
a= ------> (acceleration increasing, moving forward)
Decreasing Speed:
------------> ------> ---->
a= <------ (because it is negative)
Ticker Tape Diagrams:
- More used for numerical
- Look under Lesson 2 for more info & examples
Ticker Tape Examples:Ticker Tape vs. Motion Diagram:
- Motion Diagram has very precise measurements over ticker tape diagram
- can't see direction in ticker tape
Signs are arbitrary (made up, someone just said so)
- to change it is it allowed, but you have to label it when you do it so others know
- all you do is look at the direction in which it points and that tells you what the sign is (+ or -)
- Kinematics is the study of motion
Velocity
Lesson 2: Describing Motion with Diagrams
Introduction to Diagrams:Ticker Tape Diagrams:
Vector Diagrams:
RULE OF THUMB:
Homework Question for Lesson 2: (9/9/11)
After reading the material, answer the following questions:
1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
- I already knew about and understood a lot about the ticker tape diagram because of our lab. We talked about in class how to tell when it was speeding up or slowing down, so when reading about it I already knew about the concept. I knew about the specific spacing between each dot and what each placement represented.
- Also in class, I understood the concept of what the different arrow sizes meant on a vector diagram. I didn't have any confusion on if an object was accelerating or was at a constant speed.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.- While I was reading, I started to realize the difference between velocity and acceleration, which I was a little confused on during class. But, after reading about it and learning about the "rule of thumb" from the animation, I feel that I am not as unclear about the subject now.
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.- I think everything I read I pretty much understood. Some things I was a little confused on going into the reading, but after learning about certain subjects a second time, I believe it helped me understand everything fully.
4. What (specifically) did you read that was not gone over during class today?Class Notes for Lesson 2: (9/12/11)
Important Equations: (The Big 5)
Kinematics example problem:
Vector Diagram:
- only qualitative
ACTIVITY ON GRAPHICAL REPRESENTATIONS OF EQUILIBRIUM
9/12/11
GRAPHS:
No Motion
Constant Motion:
Fast Motion:
Slow Motion:
Discussion Questions:
1. How can you tell that there is no motion on a...
2. How can you tell that your motion is steady on a…
- position vs. time graph: slope will be steady and constant
- velocity vs. time graph: no slope, straight line
- acceleration vs. time graph: no slope, straight line
3. How can you tell that your motion is fast vs. slow on a…4. How can you tell that you changed direction on a…
5. What are the advantages of representing motion using a…
- position vs. time graph: shows direction, slope, and time intervals
- velocity vs. time graph: shows the increase or decrease of velocity from the slope
- acceleration vs. time graph: shows the change in speed from the slope
6. What are the disadvantages of representing motion using a…- position vs. time graph: doesn't specifically tell us the direction or acceleration
- velocity vs. time graph:
- acceleration vs. time graph: only shows the changes in speed
7. Define the following:GRAPH PROBLEMS:
Speed is Constant Positive, then it decreases, then it's at restNotes on Lesson 1-E: Acceleration
Acceleration: a vector quantity that is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.Homework Questions:
1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
ready knew that acceleration did not just mean an object is increasing speed. It is mainly just the rate at which an object is changing its velocity. Another thing that I already knew from class was that the direction of the acceleration vector depends on two things, whether the object is speeding up or slowing down and whether the object is moving in a positive or negative direction.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
During class, I was a little confused on the different units for average velocity. The reading helped me understand which units are for acceleration and what ones are for velocity.
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
Everything in the reading was pretty clear.
4. What (specifically) did you read that was not gone over during class today?
I don't really remember going over anything on free-falling objects. The reading explained it clearly though and now I understand that the object would have a constant acceleration due to the fact that it is falling.
Accelerating on an Incline Lab:
Danielle & Jake - 9/14/11Objectives:
- What does a position-time graph for increasing speeds look like?
- What information can be found from the graph?
Hypothesis:- A position-time graph for increasing speeds would have a constant positive slope, and form a diagonal line.
- We can find the velocity from the graph.
Available Materials:Spark tape, spark timer, track, dynamics cart, ruler/meterstick/measuring tape
Procedure:
1) Gather the materials that are needed from inside the classroom.
2) Place the ramp on top of the Physics textbook so that there is an incline.
3) Put the ticker tape through the spark timer.
4) Place the spark timer at the top of the ramp and attach the end of the ticker tape to the cart.
5) Make sure that the spark timer is on 10 Hertz then turn it on.
6) Allow the cart to drop from the top of the ramp to the bottom, forming a downward incline.
7) Remove the ticker tape from the spark timer.
8) Using a measuring device, measure the distance between each dot.
9) After recording the data, place the cart at the bottom of the ramp and then push it up.
10) Repeat steps 7 & 8.
11) Graph & analyze the recorded data on a position-time graph for both inclines.
Analysis:
Increasing Down Incline Data:
Decreasing Up Incline Data:
GRAPH:
Analysis:
a) Interpret the equation of the line (slope, y-intercept) and the R2 value.
- y= Ax2+Bx
- y= distance, x=time
- change in distance= 1/2at2 + vit
Accelerating:- Equation of the line: Δd= 15.797t2 + 6.5774t
- R2= 0.99801
Decelerating:- We found that our cart had an acceleration of 31.594 m/s because we were able to substitute the data in.
- The r2 on our graph value shows that a large amount of our data fell within 1 unit of the target point.
Instantaneous speed:
- halfway point = 61.37 m/s
- end = 46.91 m/s.
Average speed:Discussion Questions:
1. What would your graph look like if the incline had been steeper?
If the incline had been steeper, then the curve would be more dramatic and steeper. This would be due to the fact that the points would be traveling the same distance just in a shorter amount of time.
2. What would your graph look like if the cart had been decreasing up the incline?
At first, the cart would start off at a fast rate, but then gradually curve the other way. This means that the slope would start off really steep, but then decrease, and this was what caused the average to decrease. At the halfway point, more speed was gained causing a greater accelerating graph.
3. Compare the instantaneous speed at the halfway point with the average speed of the entire trip.
This makes sense because our first couple of points were unique and off to the side because of the lower speed.
At the halfway point, more speed was gained causing a greater accelerating graph. Overall, the instantaneous speed reflects a further accelerating graph, while the average speed focuses more on the slow beginning.
4. Explain why the instantaneous speed is the slope of the tangent line. In other words, why does this make sense?
The instantaneous speed is the slope of the tangent line because the line has a constant slope. Due to the fact that the original line was curved, it allowed the tangent line to hit the midway point, which allows the instantaneous speed to be estimated.
5. Draw a v-t graph of the motion of the cart. Be as quantitative as possible.
Conclusion:
In our hypothesis, we said that a position-time graph for increasing speeds would have a constant positive slope, and form a diagonal line. This part of our hypothesis was wrong because after completing the experiment and graphing the data, there slope was not constant nor always positive, so therefore it was not a diagonal line that was formed. The resulting graph turned out to be more of a curve than we expected. The correct part of our hypothesis was when we said that we would be able to find the velocity from the graph. There were many errors sources of error that were made throughout this lab. One thing is that my partner and I kept messing up our graph. We couldn't figure out why ours looked so different, because we knew that the data was collected correctly. We had to record our data again by re-doing the experiment, until the problem was finally fixed. We just realized that the axis' were wrong. Another error could have been due to the fact that our measurements were not completely accurate because we didn't have the right technology that would tell us each accurate measure. The overall goal that I got from this lab was that on a position-time graph, when the cart is accelerating then it results in a curve.
Lesson 3: Describing Motion with Position vs. Time Graphs
The Meaning of Shape for a p-t Graph:Importance of Slope on this type of graph:
Determing the Slope:
Equation:
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. In other words,
1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
During class, we talked about the difference between graphing positive and negative graphs. It was easy for me to understand while reading about it because I already grasped the concept while in class. I also already understood the shape of a position-time graph because of the motion sensor activity that we did.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
The reading helped me understand the right way to find the slope on a position time graph.
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
Everything I read was pretty clear.
4. What (specifically) did you read that was not gone over during class today?
I believe that everything I read was gone over.
Lesson 4: Describing Motion with Velocity vs. Time Graphs
The Meaning of Shape for a v-t Graph:Velocity vs. Time Graphs:
The Meaning of Slope for a v-t graph:
- In this situation above, the slope of the line is equal to the acceleration of the velocity-time graph.
- However, a car with a changing velocity will have an acceleration.
Relating the Shape to the Motion:Determining the Slope on a v-t graph:
The slope equation says that the slope of a line is found by determining the amount of rise of the line between any two points divided by the amount of run of the line between the same two points. A method for carrying out the calculation is
Determining the area on a v-t graph:
- \
Calculating the Area of a Rectangle:Calculating the Area of a Triangle:
Calculating the Area of Trapezoid:
Alternative Methods for Trapezoids:
Homework Questions:
1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully.
I already understand how to find slope from math. I also already knew how to find the area of a triangle, trapezoid, and a rectangle.
2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.
I remember us talking a little bit about the area of a shape in a v-t graph equaling the total displacement, the reading helped me understand it more than what I got out of it from the class discussion.
3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question.
Everything that I did not understand in class was pretty much covered in the reading.
4. What (specifically) did you read that was not gone over during class today?
I think that the same topics that were discussed in class were also covered in the reading.
A Crash Course in Velocity Lab (Part II)
9/21/11Partners: Jake Aronson, Jessica Smith, & Michael Solimano
Objectives:
Both algebraically and graphically, solve the following 2 problems. Then set up each situation and run trials to confirm your calculations.
Procedure:
Answers to problems:
Answer to Problem #1: 174.31 cm (where they meet)
Answer to Problem #2: 69.376 cm (where they exceed)
Car #1
- Fast
- Yellow
- CMV equation: y=28.274x
Car #2Data for Trials:
Discussion Questions:
1) Where would the cars meet if their speeds were exactly equal?
If both cars had the same speed then they would never meet. The first car would never slow down, so it would always be ahead of the second car, resulting in them never being in the same place at the same time.
2) Sketch position-time graphs to represent the catching up and crashing situations. Show the point where they are at the same place at the same time.
3) Sketch velocity-time graphs to represent the catching up situation. Is there any way to find the points when they are at the same place at the same time?
No, because on a velocity- time graph, position is not shown.
Percent Error & Percent Difference:
Crashing:
% Error:
% Difference:
Catching Up:
% Error:
%Difference:
Conclusion:
Our lab group discovered that if both a slow, yellow CMV & a fast, yellow CMV started 600 cm apart, then they would "collide" after traveling about 217.5 cm. Our theoretical data showed that the CMV's would collide 174.31 cm. When we compared this data to our actual value, our %error was 24.95%. Our theoretical data also showed that the CMV's would catch up at 69.376 cm. After completing the lab, we found out that the faster CMV caught up to the slower CMV around 65.6 cm. We compared this data to our theoretical data, and found out that we had a percent of 11.514%. One of the main sources of error that could be taken into account was the fact that the CMV's did not travel in a straight line. This helped contribute to the wide range of numbers in some of our trials. To fix this error, we could have attached both the CMV's to a straight track, so that our numbers would be more accurate. Another reason why our percent error was so high could have been because the batteries could have been different from when we last used them.
Egg Drop Project
9.28.11Partner: Dani Rubenstein
Final Project:
Description:
Our final device consisted of six cylinders holding the egg in the middle. In each of these cylinders, we placed a large amount of straws to serve as the main protective barrier for the egg. The straws also were what created the shape of the individual cylinders. We tried to build it as tight as we possible could, so therefore the egg would not fall out nor would it feel a large amount of pressure when it hit the ground.
Result:
Surprisingly after we dropped our egg it fully cracked and leaked everywhere.
Analysis:
In our first prototype, we built something that had very similar characteristics to our final one. They had the same shape, and both included a large amount of straws protecting each side. The only difference was that our first one was a little bigger, and weighed more. Even though our first protoptype was successful and kept the egg safe, we decided it would be better to minimize the size in the second prototype, so that it would weigh less. After testing out our second prototype, we learned that this shape and size was not efficient enough to keep the egg safe. In our final project, we decided to use the same design as the first one, except we cut it in half to reduce the weight. Although the first was successful, our final project did not support the egg enough causing it to crack everywhere. We learned that our project moved at a rate of 9.8 m/s2, which is very close to the average velocity of a free-falling object.
Calculations for a:
What would you do differently?
In order to make our project fully succeed, I believe that we should have built it in the shape of a cone, and placed the egg inside, surrounding it with newspaper. We also should have built some sort of parachute to slow our project down while it was in the air, and then it would have less force and impact when hitting the ground.
Lesson 5: Free Fall and the Acceleration of Gravity
Introduction to Free Fall:A free falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects:
The position of the object at regular time intervals - say, every 0.1 second - is shown. The fact that the distance that the object travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. If an object travels downward and speeds up, then its acceleration is downward.
The Acceleration of Gravity
The acceleration of gravity
- Acceleration is the rate at which an object changes its velocity. It is the ratio of velocity change to time between any two points in an object's path. To accelerate at 9.8 m/s/s means to change the velocity by 9.8 m/s each second.
If the velocity and time for a free-falling object being dropped from a position of rest were tabulated, then one would note the following pattern.
Observe that the velocity-time data above reveal that the object's velocity is changing by 9.8 m/s each consecutive second. That is, the free-falling object has an acceleration of approximately 9.8 m/s/s.
Another way to represent this acceleration of 9.8 m/s/s is to add numbers to our dot diagram that we saw earlier in this lesson. The velocity of the ball is seen to increase as depicted in the diagram at the right.
Representing Free Fall by Graphs:
A position versus time graph for a free-falling object is shown below.
A velocity versus time graph for a free-falling object is shown below.
vf = g * t
where g is the acceleration of gravity. The value for g on Earth is 9.8 m/s/s. The above equation can be used to calculate the velocity of the object after any given amount of time when dropped from rest. Example calculations for the velocity of a free-falling object after six and eight seconds are shown below.
Example Calculations:
At t = 6 s
vf = (9.8 m/s2) * (6 s) = 58.8 m/s
At t = 8 s
vf = (9.8 m/s2) * (8 s) = 78.4 m/s
The distance that a free-falling object has fallen from a position of rest is also dependent upon the time of fall. This distance can be computed by use of a formula; the distance fallen after a time of t seconds is given by the formula.
d = 0.5 * g * t2
where g is the acceleration of gravity (9.8 m/s/s on Earth). Example calculations for the distance fallen by a free-falling object after one and two seconds are shown below.
Example Calculations:
At t = 1 s
d = (0.5) * (9.8 m/s2) * (1 s)2 = 4.9 m
At t = 2 s
d = (0.5) * (9.8 m/s2) * (2 s)2 = 19.6 m
At t = 5 s
d = (0.5) * (9.8 m/s2) * (5 s)2 = 123 m
(rounded from 122.5 m)
The diagram below (not drawn to scale) shows the results of several distance calculations for a free-falling object dropped from a position of rest.
The actual explanation of why all objects accelerate at the same rate involves the concepts of force and mass. The acceleration of an object is directly proportional to force and inversely proportional to mass. Increasing force tends to increase acceleration while increasing mass tends to decrease acceleration. Thus, the greater force on more massive objects is offset by the inverse influence of greater mass. Subsequently, all objects free fall at the same rate of acceleration, regardless of their mass.
.
Falling Object Lab
10/5/11Partner: Jake Aronson
Purpose: What is the acceleration of a falling body?
Hypothesis:The acceleration due to gravity of a free-falling object is about -980 cm/s2.The v-t graph for this lab should be negative slant away from the origin. We will find the "g" value from the equation Ax2+bx, where the "A" will represent the "g" value.
Our Data:
Class Data:
Analysis & Graphs:
Position-Time Graph:
From this graph, we can see that our acceleration is 756.56 cm.s2. This is because the slope of our trend line is equal to the acceleration.We can calculate the acceleration by using the equation in which this graph uses, which is y=Ax2+Bx. The formula that we used in class for this graph was, y=vit+½at2. Being that the r2 is so close to 1 here, our data is almost perfect when compared to the trend line.
Velocity-Time Graph:
This graph is graphed in the form of y=mx+b, or an equation of a line. The equation that we used in class for this graph is
vf=vi+at. The difference between these two is that instead of multiplying the slope like we did in the first one, here we already have the acceleration given to us, 755.87 cm/s2. Just like in the position time graph above, our r2 value is very close to 1, so again we have an almost perfect match between our data and trend line.
Sample Calculations:
Discussion Questions:
1. Does the shape of your vt graph agree with the expected graph? Why or why not?
No, because on the expected graph the velocity line to be negative considering we were dropping something. But, the actually graph showed the complete opposite. This was because instead of plugging in the negative data, we used the positive data. On the expected graph, the velocity line was suppose to be on a negative slant away from the origin. Our graph showed a positive line, so they were completely different.
2. Does the shape of your xt graph agree with the expected graph? Why or why not?
Yes, both graphs agree with each other. Both graphs have a J-curve because of the increasing velocity and acceleration.
3. How do you results compare to that of the class? (Use percent difference to discuss quantitatively).
After graphing our data, we discovered that our slope was 755.28, and the class average was 839.417. After finding out both these numbers, we found that the %difference was 10.02, so we had about a 10% difference from the class.
4. Did the object acceleration uniformly? How do you know?
Yes, because in the v-t graph there is a constant slope.
5. What factor(s) would cause acceleration due to gravity to be higher than it should be? Lower than it should be?
One factor that could have caused acceleration due to gravity to be higher could have been the placement of our hands while we were dropping the weights. A factor that could have either cause the gravity to be higher or lower could have just been measurements because the tape was not sturdy the whole time.
Conclusion:
Our data turned out to be very accurate and our hypothesis was not fully correct because of the fact that we predicted to have a negative velocity, but the graph turned out to have a positive slope. We thought that it would be negative because we were dropping something. Some causes of error that could have caused our results to be a little off form others could have been the friction that was caused from the ticker tape rubbing against the ticker tape timer. Another issue was when we were measuring our data. We could have made a mistake while doing this because the tape was not staying on the ground, and would keep coming up. To fix this, we could have been more cautious and taped it better so that it would be less likely to shift. We also could have dropped the weights from something other than our hands, so that no extra momentum would be caught.